Application of Comsol Multiphysics 3.2 to finite strain ... - Humusoft

Application of Comsol Multiphysics 3.2 to finite strain viscoelasticity of anelastomeric solid.Bohdana Marvalova *AbstractThe paper presents an implementation in Comsol Multiphysics 3.2 of Holzapfel's material model forthe viscoelastic stress response of carbon-black filled rubber at large strains. The simulation in Comsol3.2 uses the Structural Mechanics Module finite strain formulation based on the material configurationwith the right Cauchy-Green tensor as a strain measure. The time independent response of the rubber ismodelled by the Mooney-Rivlin hyperelastic material with uncoupled volumetric/deviatoric free energyfunction. In addition to the volumetric and isochoric elastic response function we use a configurationfree energy, which drives the viscoelastic response. Thus we obtain a decoupled stress response whichconsists of equilibrium and non-equilibrium parts. The non-equilibrium viscoelastic stresses areevolving internal variables governed by rate equations which are modelled in PDE General Form. Inour simulation we used some predefined SME variables and we defined several other suitable variablessuch as the isochoric elastic stresses and material tangent modulus. The paper presents threesimulations of the elastomeric viscoelastic solids response in relaxation, creep and cyclic loading.KeywordsRubber behaviour, Viscoelasticity , Large strains, Stress relaxation, Cyclic loading1. IntroductionRubber materials are applied in various branches of mechanical engineering because of theirdamping properties. The modelling and FEM calculation of the structural response requires aconstitutive model which captures the complex material behaviour. The present paper focuses on theviscoelastic behaviour of the filled rubber.The ground-stress response of filled rubber is usually modelled in the phenomenologicalframework of finite elasticity by Mooney-Rivlin or Ogden models, or by Aruda and Boyce model interms of the micromechanically based kinetic theory of polymer chain deformations.Beside the elastic response the filled rubber shows also the finite viscoelastic overstressresponse which is apparent in creep and relaxation tests. Cyclic loading tests show a typical frequencydependenthysteresis as well where the width of the hysteresis increases with increasing stretch rates.The constitutive theory of finite linear viscoelasticity is a major foundation for modeling ratedependentmaterial behaviour based on the phenomenological approach. In this approach, a suitablehyperelasticity model is employed to reproduce the elastic responses represented by the springs, whilethe dashpot represents the inelastic or the so-called internal strain. Its temporal behavior is determinedby an evolution equation.2. Model for finite viscoelasticityThe material model of finite strain viscoelasticity used in our work follows from the concept ofSimo [2] and Govindjee & Simo [4]. The finite element formulation of the model was elaborated by* Bohdana Marvalova, Dept. of Engineering Mechanics, Technical University of Liberec, Halkova 6, 46117 Liberec,Czech Republic, bohdana.marvalova@seznam.cz

Holzapfel [3] and used by Holzapfel & Gasser [5] to calculate the viscoelastic deformation of fiberreinforced composite material undergoing finite strains.The model is based on the theory of compressible hyperelasticity with the decoupledrepresentation of the Helmholtz free energy function with the internal variables (Holzapfel [1], p. 283) :C , 1,..... , m= ∞VOL J ∞ISOmC ∑ C , , C= J −2/3 C . (1)The first two terms in (1) characterize the equilibrium state and describe the volumetric elastic responseand the isochoric elastic response as t ∞ , respectively. The third term is the dissipative potentialresponsible for the viscoelastic contribution. The derivation of the 2 nd Piola-Kirchhoff stress withvolumetric and isochoric terms:∞S VOL∞S ISOS=2 ∂ C , 1 , ..... , m ∂C=1=S ∞VOLS ∞ISOm∑=1Q (2)where and is the volumetric and the isochoric stress response respectively and theoverstress Q is stress of 2 nd Piola-Kirchhoff type.S ∞ISOS ∞VOL=J d ∞VOLd J J C −1 ,=J −2/ 3 Dev [2 ∂ ∞ISO∂ CC ] ,Dev .=.−1/3[ .:C ]C −1 . (5)(3)(4)where Dev . is the deviatoric operator in the Lagrangian description. Motivated by the generalizedMaxwell rheological model (Fig. 1), the evolution equation for the internal variable Q takes on theform (6).Fig. 1. Maxwell rheological modelṠ ∞ISOQ˙ Q =Ṡ ISO,Ṡ ISO= ∞Ṡ ∞ ISOC ,whereC = ∂ S ∞ISO∂ C Ċ =C Ė ISOC ISO=2 ∂ S ∞ISO∞∂ C =2 ∂ S ISO∂C∂C∂C(6)(7)(8)(9) ∞ ∈0,∞ in the expression (7) is the nondimensional strain energy factor [2,4], is therelaxation time, C ISO is isochoric contribution of the tangent elasticity tensor and Ė is the materialstrain rate tensor

Ė= 1 2 ˙ F T FF T Ḟ .(10)The material is assumed slightly compressible, the volumetric and isochoric (Mooney - Rivlin) parts ofHelmholtz free energy function were chosen in the form ∞ VOL J = 1 d J −12 , ∞ ISOC =c 1 I 1−3c 2 I 2−3, (11)with the parameters c 1,c 2 and d. The viscoelastic behavior is modeled by use of =2 relaxation∞processes with the corresponding relaxation times and free energy factors . All theparameters were determined from experimental measurements.3. Finite element simulation in Comsol MultiphysicsThe material model described above was implemented into Comsol Multiphysics. TheStructural Mechanics and PDE modules were used for the calculation of time dependent response of arubber block in plain strain in different loading regimes. The implementation is very similar to theViscoelastic material case in Structural Mechanics Model Library. The application mode type planestrain in Structural Mechanics Module, the time dependent analysis and the Mooney-Rivlinhyperelastic material were chosen. The components of the isochoric stress rate were determinedin Symbolic Toolbox in Matlab and added to the scalar expression table in Comsol. PDE module wasused for the integration of the evolution equation (6). The results of the different simulations show thegood qualitative agreement with experimental time dependent behaviour of filled rubbers. The similarmodel based on Prony series was implemented into ANSYS 10 [6] and the results of simulations arecomparable with our ones.∞Ṡ ISOFig. 2. Time dependent displacement controled loading of rubber block

Fig. 3. Creep testFig. 4. Relaxation test4. ConclusionThe paper has presented the FEM implementation in Comsol Multiphysics of a viscoelasticmaterial model in finite strain in the Lagrangian configuration. Simple examples were subsequentlypresented, namely the relaxation, creep and time dependent loading of a rubber block. The block wasmodelled using Structural Mechanics and PDE modules.AcknowledgementThe author gratefully acknowledges the funding provided by the Czech Grant Agency GACR – grantNo.101/05/2669.References[1] Holzapfel, G. A., Nonlinear solid mechanics. Wiley, 2000., ISBN 0471823198[2] Simo, J.C., On a fully three dimensional finite strain viscoelastic damage model: formulation andcomputational aspects. Comput. Meth. Appl. Mech. Eng. 60 (1987), 153–173.[3] Holzapfel, G.A., On large strain viscoelasticity: continuum formulation and finite elementapplications to elastomeric structures. Int. J. Numer. Meth. Eng. 39 (1996), 3903–3926.[4] Govindjee, S., Simo, J.C., Mullins' effect and strain amplitude dependence of the storage modulus.Int. J. Solid Struct. 29 (1992), 1737–1751[5] Holzapfel, G. A., Gasser, T.C., A viscoelastic model for fiber-reinforced composites at finite strains:Continuum basis, computational aspects and applications. Computer Methods in Applied Mechanicsand Engineering Vol. 190 (2001) 4379-4430[6] Dokumentation for ANSYS 10, verification manual, VM 234